New Directions in Geometric Group Theory and Topology

这个项目的目标是对所有级别的数学产生最大的影响。在研究层面，该项目引入了新的几何观点来理解4维空间。每当有4个参数出现时，这些空间就会出现，例如跟踪飞机的三维位置以及它在该位置的时间。该项目将开发新的方法来理解这些空间的对称性，从物理学中出现的一个基本例子--“K3流形”开始。该项目还将导致对几名博士生的培训；继续国际数学联合会在媒体上的公开宣传；并通过继续国际数学联盟的工作，促进数学多样性的工作。国际数学联盟建议将瑟斯顿观点引入到绘制K3表面的类群的研究中。具体目标包括：瑟斯顿三分法；瑟斯顿范式；尼尔森(非)实现；新的美丽的例子/结构。在第二个方向上，PI提出了一个指导原则，该指导原则受到代数几何奇迹般的构造的启发，在几何群论、拓扑学和代数几何中产生了许多自然的、有动机的问题和猜想，涉及许多不同模空间的各种形式的刚性。作为更广泛影响的一部分，PI还将培训研究生并进行推广。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The goal of this project is to have maximal impact on mathematics at all levels. At the research level, the project introduces new geometric points of view to our understanding of 4-dimensional spaces. These spaces arise whenever 4 parameters arise, for example keeping track of the 3-dimensional position of an airplane together with the time it is at that position. The project will develop new methods to understand symmetries of these spaces, starting with ``K3 manifolds'', a fundamental example arising in physics. The project will also lead to the training of several PhD students; continue the PI's public outreach in the media; and contribute to work for diversity in mathematics by continuing the PI's work with the Math Alliance.The PI proposes to introduce the Thurstonian point-of-view into the study of mapping class groups of K3 surfaces. Specific goals include: Thurston trichotomy; Thurston normal form; Nielsen (non)realization; new beautiful examples/constructions. In a second direction, the PI proposes a guiding principle, motivated by the miraculous constructions of algebraic geometry, that produces many natural, motivated problems and conjectures in geometric group theory, topology and algebraic geometry, concerning various forms of rigidity for many different moduli spaces. The PI will also train graduate students and conduct outreach as part of broader impacts.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.